We give complexity analysis of the class of short generating functions (GF).
Assuming #P is not in FP/poly, we show that this class is
not closed under taking many intersections, unions or projections of GFs,
in the sense that these operations can increase the bit length of coefficients
of GFs by a super-polynomial factor. We also prove that
truncated theta functions are hard in this class.

We prove that integer programming with three quantifier alternations is NP-complete, even for a fixed number of variables.
This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two
quantifier alternations can be done in polynomial time for a fixed number of variables. As a byproduct of the proof,
we show that for two polytopes P,Q⊂R4, counting the projection of integer points in Q\P is #P-complete.
This contrasts the 2003 result by Barvinok and Woods, which allows counting in polynomial time the projection of
integer points in P and Q separately.

We study complexity of short sentences in Presburger arithmetic (Short-PA).
Here by "short'' we mean sentences with a bounded number of variables, quantifiers,
inequalities and Boolean operations; the input consists only of the integers involved
in the inequalities. We prove that assuming Kannan's partition can be found in
polynomial time, the satisfiability of Short-PA sentences can be decided
in polynomial time. Furthermore, under the same assumption, we show that
the numbers of satisfying assignments of short Presburger sentences
can also be computed in polynomial time.

We extend the Barvinok-Woods algorithm for enumeration of integer points in
projections of polytopes to unbounded polyhedra. For this, we obtain a new
structural result on projections of semilinear subsets of the integer lattice.
We extend the results to general formulas in
Presburger Arithmetic. We also
give an application to the k-Frobenius problem.

We investigate the problem of sampling integer points in rational
polyhedra provided an oracle for counting these integer points.
When the dimension is bounded, this assumption is justified in view of a
recent algorithm
due to Barvinok. We show that in
full generality the exactly uniform sampling is possible, when the
oracle is called polynomial number of times. Further,
when Barvinok's algorithm is used, poly-log number of calls
suffices.

We give a new proof of Steinitz's classical theorem in the case of plane triangulations,
which allows us to obtain a new general bound on the grid size of the simplicial polytope
realizing a given triangulation, subexponential in a number of special cases.
Formally, we prove that every plane triangulation G with n vertices can be
embedded in R2 in such a way that it is the vertical projection of a convex polyhedral
surface. We show that the vertices of this surface may be placed in a
4n3 x 8n5 x ζ(n) integer grid, where ζ(n) < (500 n8)τ(G) and
τ(G) denotes the shedding diameter of G, a quantity defined in the paper.

The main result in the paper is a construction of a simple (in fact, just a union of two
squares) set T in the plane with the following property. For every ε> 0 there is a family F of
an odd number of translates of T such that the area of those points in the plane that belong to
an odd number of sets in F is smaller than ε.

In the Graph Realization Problem (GRP), one is given a graph G, a set of non-negative edge-weights, and an integer d. The goal is to find, if possible, a realization of G in the Euclidian space Rn, such that the distance between any two vertices is the assigned edge weight. The problem has many applications in mathematics and computer science, but is NP-hard when the dimension d is fixed. Characterizing tractable instances of GRP is a classical problem, first studied by Menger in 1931 in the case of a complete graph. We construct two new infinite families of GRP instances whose solution can be approximated up to an arbitrary precision in polynomial time. Both constructions are based on the blow-up of fixed small graphs with large expanders. Our main tool is the Connelly's condition in Rigidity Theory, combined with an explicit construction and algebraic calculations of the rigidity (stress) matrix. As an application of our results, we give a deterministic construction of uniquely
k-colorable vertex-transitive expanders.

Download .pdf file of the paper.
Published online by DCG September 25, 2013 here.

We study the problem of acute triangulations of convex polyhedra
and the space Rn. Here the acute triangulation is a triangulation
into simplices whose dihedral angles are acute. We prove that acute
triangulations of the n-cube do not exist for n≥4. Further,
we prove that acute triangulations of the space Rn do not exist
for n≥5. In the opposite direction, we present a construction
of an acute triangulation of the cube and the regular octahedron in R3.
We also prove nonexistence of an acute triangulation of R4 if all
dihedral angles are bounded away from π/2.

Download .pdf file of the paper,
or .pdf of the French abstract.
Go to
this page
to see animations of acute triangulations from the paper.

The extended abstract of the paper has appeared in
Proc. Symp. Computational Geometry
(SCG'2010), ACM, New York, 2010, 307–313.
Download .pdf file of the extended abstract.
Note that the proceedings version does not contain the Appendix.

(with Rom Pinchasi)
How to cut out a convex polyhedron, preprint (2011).

It is known that one can fold a convex polyhedron from a
non-overlapping face unfolding, but the complexity of the
algorithm in [Miller-Pak] remains an open problem.
In this paper we show that every convex polyhedron P
in Rd
can be obtained in polynomial time, by starting with a cube which
contains P and sequentially cutting out the extra parts
of the surface.

Our main tool is of independent interest. We prove that
given a convex polytope P in Rd
and a facet F of P,
then F is contained in the union of AG.
Here the union is over all the facets G of P
different from F,
and AG is the set obtained from G
by rotating towards F the
hyperplane spanned by G about the intersection of it with the
hyperplane spanned by F.

We study the shape of inflated surfaces introduced in
[Bleecker]
and [Pak]. More precisely, we analyze
profiles of surfaces obtained by inflating a convex polyhedron,
or more generally an almost everywhere flat surface, with a
symmetry plane. We show that such profiles are in a
one-parameter family of curves which we describe explicitly
as the solutions of a certain differential equation.

We prove that all polyhedral surfaces in R3
have volume-increasing piecewise-linear isometric deformations.
This resolves the conjecture of Bleecker who proved it for
convex simplicial surfaces
[Bleecker].
Further, we prove that all convex polyhedral surfaces in Rd
have convex volume-increasing piecewise-linear
isometric deformations. We also discuss the limits on
the volume of such deformations, present a number of
conjectures and special cases.

Download .pdf file (676 K) of the paper.
The pictures are better viewed in color, but should print fine
on a monochromatic printer.

In his works
[R1]
and [R2]
David Robbins proposed several
interrelated conjectures on the area of the polygons inscribed
in a circle as an algebraic function of its sides. Most recently,
these conjectures have been established in the course of
several independent investigations. In this note we give an
informal outline of these developments.

We prove that for every two d-dimensional
convex polytopes P, Q with vol(P)= vol(Q),
there exists a continuous piecewise-linear (PL) volume-preserving
map f: P->Q. The result extends to
general PL-manifolds. The proof is inexplicit and uses
the corresponding fact in the smooth category, proved by
Moser[M].
We conclude with various examples and
combinatorial applications.

We present an algebraic approach to the classical problem of
constructing a simplicial convex polytope given its planar
triangulation and lengths of its edges. We introduce a ring of
polynomial invariants of the polytope and show that they
satisfy polynomial relations in terms of squares of edge lengths.
We obtain sharp upper and lower bounds on the degree of these
polynomial relations. In a special case of regular
bipyramid we obtain explicit formulae for some of these
relations. We conclude with a proof of Robbins
Conjecture on the degree of generalized Heron
polynomials.

We initiate a systematic investigation
of the metric combinatorics of convex polyhedra by proving
the existence of polyhedral nonoverlapping unfoldings and analyzing
the structure of the cut locus. The algorithmic aspect, which we
include together with its complexity analysis, was for us a
motivating feature of these results. We also show
that our general methods extend to the abstract spaces we call
'convex polyhedral pseudomanifolds', whose sectional curvatures
along low-dimensional faces are all positive. We also
propose some directions for future research, including a series of
precise conjectures on the number of combinatorial types of
shortest paths, and on the geometry of unfolding boundaries of
polyhedra.

Download
.pdf file of the paper.
The pictures are colored and better viewed in color.
They should print fine on a monochromatic printer.

On the number of faces of certain transportation polytopes,
European J. Combinatorics, vol. 21 (2000), 689-694.

Define transportation polytope T(n,m) to be a
polytope of nonnegative n x m matrices with row sums
equal to m and column sums equal to n.
We present an efficient algorithm for computing
the numbers of the k-dimensional faces for the
transportation polytope T(n,n+1).
The construction relies on the new recurrence relation for
which is of independent interest.

Domino tileability is a classical problem in Discrete Geometry,
famously solved by Thurston for simply connected regions in nearly
linear time in the area. In this paper, we improve upon Thurston's
height function approach to a nearly linear time in the perimeter.

Tiling planar regions with dominoes is a classical problem, where the decision
and counting problems are polynomial. We establish a variety of hardness results
(both NP- and #P-completeness), for different generalizations of dominoes in
three and higher dimensions.

Download .pdf file of the paper (best printed in color).
Link to the journal version: is here.

In 1995, Beauquier, Nivat, Rémila, and Robson showed
that tiling of general regions with two rectangles is NP-complete,
except for few trivial special cases. In a different direction, in 2005, Rémila
showed that for simply connected regions and two rectangles, the tileability
can be solved in quadratic time (in the area). We prove that there is a finite
set of at most 106 rectangles for which the tileability problem of
simply connected regions is NP-complete, closing the gap between positive
and negative results in the field. We also prove that counting such
rectangular tilings is #P-complete, a first result of this kind.

Download .pdf file of the paper (Warning: it is color optimized,
so if printed try to use a color printer).
See also my blog post inspired by this paper.

Note: The above version is identical to the arXiv version.
The journal version: .pdf file.
It is somewhat abridged at journal's request. Notably, it is missing proof of Lemma 3.2 replaced with a UTM argument.
Jed Yang's thesis (2012) contains
expanded definitive versions of both, with constants substantially improved.

We prove that any two tilings of a rectangular region by
T-tetrominoes
are connected by moves involving only 2
and 4 tiles. We also show that the number of such
tilings is an evaluation of the
Tutte polynomial.
The results are extended to more general class of regions.

Let T be a finite set of tiles. The group of invariants
G(T), introduced by the author, is a group of linear
relations between the number of copies of the tiles in tilings of the same
region. We survey known results about G(T), the height function approach,
the local move property, various applications and special cases.

Ribbon tiles are polyominoes consisting of n squares laid out in a
path, each step of which goes north or east. Tile invariants were
first introduced in [Pak], "Ribbon tile invariants" (see below),
where a full basis of invariants of
ribbon tiles was conjectured. Here we present a complete proof of the
conjecture, which works by associating ribbon tiles with a certain
polygon in the complex plane, and deriving invariants from the signed
area of this polygon.

Consider a set of ribbon tiles which are polyominoes
with n squares obtained by up and right rook moves.
We describe all the linear relations for the number of times each
such tile can appear in a tiling of any given row convex region.
We also investigate the connection with signed tilings and give
applications of tileability.

Let F ⊂ Sk be a finite set of permutations and
let Cn(F) denote the number of permutations
σ ∈ Sn avoiding the set of patterns F.
The Noonan-Zeilberger conjecture states that the sequence
{Cn(F)} is P-recursive. We disprove this
conjecture. The proof uses Computability Theory and builds on our
earlier work [GP1]. We also give two applications
of our approach to complexity of computing {Cn(F)}.

(with Scott Garrabrant)
Words in linear groups, random walks, automata and P-recursiveness, preprint (2015);
to appear in the inaugural issue of the Journal of Combinatorial Algebra.

Let S be a generating set of a finitely generated group G .
Denote by an the number of words in S of length n
that are equal to 1. We show that the cogrowth sequence {an} is not
always P-recursive. This is done by developing new combinatorial
tools and using known results in computability and probability on groups.

Download the .pdf file of the paper.
Watch my talk
on the paper (BIRS, March 9, 2015).

We introduce and study the number of tilings of unit height
rectangles with irrational tiles. We prove that the class of
sequences of these numbers coincides with the class of
diagonals ofN-rational generating functions
and a class of certain binomial multisums.
We then give asymptotic applications and establish
connections to hypergeometric functions and
Catalan numbers.

Download the .pdf file of the paper
(best printed on a color printer).

Guibert and Linusson introduced in [GL] the family of doubly alternating Baxter permutations,
i.e. Baxter permutations σ in Sn, such that σ
and σ-1 are alternating. They proved
that the number of such permutations in S2n and
S2n+1 is the Catalan numberCn.
In this paper we explore the expected limit shape of such permutations,
following the approach by Miner and Pak [MP].

Download the .pdf file of the paper (best printed on a color printer).
Here is the journal page for the paper.
Warning: the file is 1.8Mb.

We initiate the study of limit shapes for random permutations avoiding
a given pattern.
Specifically, for patterns
of length 3, we obtain delicate results on the asymptotics of distributions of positions of numbers in
the permutations. We view the permutations as 0-1 matrices to
describe the resulting asymptotics geometrically. We then apply our
results to obtain a number of results on distributions
of permutation statistics.

Download the .pdf file of the paper (best printed on a color printer).

Note: At the request of editors, the preprint version has been shortened for publication.
The published article is missing some pictures, few corollaries and some proof details. Here is the
.pdf file of the article.

In 1857, Cayley
showed that certain sequences, now called Cayley compositions,
are equinumerous with certain partitions into powers of 2. In this paper we
give a simple bijective proof of this result and obtain several extensions.
We then extend this bijection to an affine linear map between convex polyhedra to
give and new proof of Braun's conjecture.

Cayley polytopes were defined recently
as convex hulls of Cayley compositionsintroduced
by Cayley in 1857. In this paper we resolve Braun's conjecture,
which expresses the volume of Cayley polytopes in terms of the number of connected graphs.
We extend this result to two one-variable deformations of Cayley polytopes (which we call
t-Cayley and t-Gayley polytopes), and to the most general two-variable deformations,
which we call Tutte polytopes. The volume of the latter is given via an evaluation of the
Tutte polynomial of the complete graph.

Our approach is based on an explicit triangulation of the Cayley and Tutte polytope.
We prove that simplices in the triangulations correspond to labeled trees,
and use properties of the Tutte polynomials to computes the volume of the polytopes.
The heart of the proof is a direct bijection based on the
neighbors-first searchgraph traversal algorithm.

An abc-permutation is a permutation
σabc in Sn
generated by exchanging an initial block of length a and
a final block of length c of {1...n}, where n=a+b+c.
In this note we compute the limit of the probability that a
random abc-permutation is a long cycle. This resolves an
open problem V. Arnoldposed in 2002.

The published version contained a mistake in the proof of Lemma 2.
The mistake was fixed in the erratum and the new proof of Lemma 2
is much shorter and simpler.

We present a new algebraic extension of the classical
MacMahon Master Theorem. The basis of our extension
is the Koszul duality for non-quadratic algebras defined
by Berger.
Combinatorial implications are also discussed.

We present several non-commutative extensions of the MacMahon
Master Theorem, further extending the results of Cartier-Foata
and Garoufalidis-Lê-Zeilberger. The proofs are combinatorial
and new even in the classical cases. We also give applications
to the β-extension and Krattenthaler-Schlosser's
q-analogue.

Download
.pdf file of the paper.
See also a .pdf file of an
iextended abstract (11 pages), which will appear in the
FPSAC'07
Conference Proceedings.

We present a bijection between 321- and 132-avoiding permutations
that preserves the number of fixed points and the number of
excedances. This gives a simple combinatorial proof of recent
results of Robertson, Saracino and Zeilberger
[RSZ],
and the first author [E].
We also show that our bijection preserves additional statistics,
which extends the previous results.

We consider several classes of increasing trees, which are
equinumerable
with alternating
permutations (also called updown permutations).
We also consider various statistics on these trees and relations with
André polynomials and the Foata group, Entringer and
Euler-Bernoulli
numbers. Most proofs are bijective.

We give new product formulas for the number of standard Young tableaux
of certain skew shapes and for the principal evaluation of the
certain Schubert polynomials. These are proved by utilizing symmetries
for evaluations of factorial Schur functions, extensively studied in
the first two papers in the series [MPP1,MPP2]. We also apply
our technology to obtain determinantal and product formulas for the
partition function of certain weighted lozenge tilings, and give
various probabilistic and asymptotic applications.

Download .pdf file of the paper. Warning: the pictures are nicely colored, so the paper is best read
when printed on a color printer. Welcome to the 21st century!

(with Alejandro Morales and Greta Panova)
Asymptotics of the number of standard Young tableaux of skew shape, preprint (2016), 23 pp.

We give new bounds and asymptotic estimates on the number of
standard Young tableaux of skew shape in a variety of special
cases. Our approach is based on Naruse's hook-length formula.
We also compare our bounds with the existing bounds on the numbers
of linear extensions of the corresponding posets.

The Naruse hook-length formula is a recent general formula for
the number of standard Young tableaux of skew shapes, given as a positive
sum over excited diagrams of products of hook-lengths. In
[MPP1] we gave two different q-analogues of Naruse's formula: for the
skew Schur functions, and for counting reverse plane partitions of skew
shapes. In this paper we give an elementary proof of Naruse's formula
based on the case of border strips. For special border strips, we obtain curious
new formulas for the Euler and q-Euler numbers in
terms of certain Dyck path summations.

The celebrated hook-length formula gives a product formula
for the number of standard Young tableaux of a straight shape.
In 2014, Naruse announced a more general formula for
the number of standard Young tableaux of skew shapes as a positive
sum over excited diagrams of products of hook-lengths.
We give an algebraic and a combinatorial proof of Naruse's formula,
by using factorial Schur functions and a generalization of the
Hillman-Grassl correspondence, respectively.

The main new results are two different q-analogues of Naruse's formula for the
skew Schur functions and for counting reverse plane partitions of skew
shapes. We establish explicit bijections between these objects and
families of integer arrays with certain nonzero entries, which also
proves the second formula.

Download .pdf file of the paper.
Warning: the pictures are colored, so the paper is best read
when printed on a color printer.

Note: The original version of the paper has been expanded
and split into two papers in a series: I and II.
Download .pdf file of the original version (44 pp.)

Supplemental materials for this and other paper in the series:
1) Sage worksheet with most examples and conjectures in the paper,
2) an elementary but curious special case, see also a
direct proof on Math.StackExchange, and WolframAlpha check of the key
induction step.

We present a lower bound on the Kronecker coefficients of the symmetric group via the
characters of Sn, which we apply to obtain various explicit estimates.
Notably, we extend Sylvester's unimodality of q-binomial coefficients as
polynomials in q to derive sharp bounds on the differences of their consecutive
coefficients. We then derive effective asymptotic lower bounds for a wider class of
Kronecker coefficients.

We study the complexity of computing Kronecker coefficientsg(α,μ,ν). We give explicit bounds in terms of the number of
part ℓ in the partitions, their largest part size N and the
smallest second part M of the three partitions. When M = O(1),
i.e. one of the partitions is hook-like, the bounds are linear
in log N, but depend exponentially on ℓ.
Moreover, similar bounds hold even when M=exp O(ℓ).
By a separate argument, we show that the positivity of Kronecker
coefficients can be decided in O(log N) time for a bounded
number ℓ of parts and without restriction on M.
Related problems of computing Kronecker coefficients when one partition
is a hook, and computing characters of Sn
are also considered.

We prove strict unimodality of the q-binomial (Gaussian) coefficients as
polynomials in q. The proof is based on the combinatorics of certain
Young tableaux and the semigroup property of Kronecker coefficients
of Sn representations. This extends classical unimodality
result by J.J. Sylvester.

Download .pdf file of the paper.
Note: This version is a bit different from the published article,
revised to fix a subtle mistake. It is the same as latest arXiv version.
Here is Sylvester's
original article"Proof of the hitherto undemonstrated Fundamental Theorem of Invariants"
(Philosophical Magazine, 1878).
Here is Cayley's
original article"A Second Memoir upon Quantics" (Philosophical Transactions of the Royal Society of London, 1856)
with the unimodality conjecture.

We present new proofs and generalizations of unimodality of the
q-binomial (Gaussian) coefficients as polynomials in q.
We use an algebraic approach by interpreting the differences between numbers
of certain partitions as Kronecker coefficients of representations of Sn.
Other applications of this approach include strict unimodality of the diagonal
q-binomial coefficients and unimodality of certain partition statistics.

We study the remarkable Saxl conjecture which states that tensor squares of
certain irreducible representations of the symmetric groups Sn
contain all irreducibles as their constituents. Our main result is that they contain
representations corresponding to hooks and two row Young diagrams.
For that, we develop a new sufficient condition for the
positivity of Kronecker coefficients in terms of characters,
and use combinatorics of rim hook tableaux combined with known results
on unimodality of certain partition functions. We also present connections
and speculations on random characters of Sn.

Following the work of Okounkov-Pandharipande [OP1],
[OP2] and
Diaconescu [D],
we study the equivariant quantum cohomology of the
Hilbert scheme and the relative Donaldson-Thomas theory.
Using the ADHM construction
and a recent work [CDKM], we continue the study the
Gromov-Witten invariant of the
abelian/nonabelian correspondence, and establish a connection
between the J-function
of the Hilbert scheme and a certain combinatorial identity in two variables.
This identity is then generalized to a multivariate identity, which
simultaneously generalizes the branching rule for the dimension of
irreducible representations of the symmetric group in the staircase
shape. We then establish this identity by a weighted generalization
of the
Greene-Nijenhuis-Wilf hook walk,
which is of independent interest.

Based on the ideas in our previous paper [CKP], we introduce the weighted analogue
of the branching rule for the classical hook length formula,
and give two proofs of this result. The first proof is completely
bijective, and in a special case gives a new short combinatorial
proof of the hook length formula. Our second proof is probabilistic,
generalizing the (usual) hook walk proof of Green-Nijenhuis-Wilf,
as well as the q-walk of Kerov. Further applications are also presented.

We introduce notions of linear reduction and linear equivalence
of bijections for the purposes of study bijections between Young
tableaux. Originating in Theoretical Computer Science, these
notions allow us to give a unified view of a number of classical
bijections, and establish formal connections between them.
Along the way we establish a number unexpected connections
between classical Young tableaux
bijections and make several intriguing conjectures.

We present several direct bijections between different
combinatorial interpretations of the
Littlewood-Richardson
coefficients. The bijections are defined by explicit
linear maps which have other applications.

We present a transparent proof of the classical hook length
formula. The formula is reduced to an equality between the number of
integer point in certain polytopes. The latter is established by
an explicit continuous volume-preserving piecewise linear map.

Download .pdf file of the paper.
Click
here for a journal version. See also
here for a nice web version of the idea.

You can also check
the
Third Edition of
Don Knuth's
The
Art of Computer Programming (Vol. 3, 1998),
for a 2-page overview of the algorithm. Yet another
2-page overview is in Bruce Sagan's "Group Representations and
Symmetric Functions", MSRI Lecture Notes, 1997 (available
here) and
the new edition
of his monograph The Symmetric Group (which calls the bijection
"beautiful", rightly or not).

Finally, to answer a common question: yes, the
shifted analogue
of the bijection is now known,
and due to Ilse Fischer
(in our paper we announced this possibility, but never wrote a paper since we
could not figure out all the technical details).

We present several remarkable properties of the one
representation of Sn, obtained by an action on parking
functions.
Of particular importance are multiplicities of the irreducible
representations corresponding to hook shapes which correspond
to certain k-trees.

We construct a new resolution for a special type of
Sn-modules.
The resolution arises from inversion polynomial and generalizes
a known combinatorial identity. In the limiting case we obtain
new and classical partition identities.

A classical hook-content formula appears as a Poincare series
for the multiplicities of the irreducible
Sn-module in symmetric algebra.
We obtain a super-analog of this formula by taking
Weil algebra instead of the symmetric algebra.

We compute the limit shape for several classes of restricted integer partitions,
where the restrictions are placed on the part sizes rather than the multiplicities.
Our approach utilizes certain classes of bijections which map limit shapes continuously
in the plane. We start with bijections outlined in [Pak], and extend them to include
limit shapes with different scaling functions.

We prove that the partition function p(n) is log-concave for all n>25.
We then extend the results to resolve two related conjectures by Chen and one by Sun.
The proofs are based Lehmer's estimates on the remainders of the Hardy-Ramanujan
and the Rademacher series for p(n).

Acknowledgement of priority: After the paper was published, we learned
that the central result of the paper has appeared earlier in the following interesting article.
The proofs are different, but of the same flavor, although our version is more
detailed and has other applications.
Jean-Louis Nicolas, Sur les entiers N pour lesquels il y a beaucoup
de groupes abéliens d'ordre N (in French), Annales de l’institut Fourier, tome 28, no. 4 (1978), p. 1-16.

In this paper we analyze O'Hara's partition bijection. We present three type of results.
First, we show that O'Hara's bijection can be viewed geometrically as a certain scissor
congruence type result. Second, we obtain a number of new complexity bounds,
proving that O'Hara's bijection is efficient in several special cases and mildly
exponential in general. Finally, we prove that for identities with finite support,
the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more
efficiently than by O'Hara's construction.

We introduce a notion of asymptotic stability for bijections
between sets of partitions and a class of geometric bijections.
We then show that a number of classical partition bijections
are geometric and that geometric bijections under certain
conditions are asymptotically stable.

Download .pdf file (500K) of the paper.
Note that the pictures are colored, so the paper is best read
when printed on a color printer.

We analyze involutions which prove several partition identities and
describe them in a uniform fashion as projections of "natural"
partition involutions along certain bijections.
The involutions include those due to Franklin, Sylvester, Andrews,
as well as few others.
A new involution is constructed for an
identity of Ramanujan,
and analyzed in the same fashion.

We present an extensive survey of bijective proofs of classical
partitions identities. While most bijections are known, they
are often presented in a different, sometimes unrecognizable way.
Various extensions and generalizations are added in the form of
exercises.

In the MathSciNet review,
George Andrews writes:
"This paper undertakes a monumental task: to present a reasonably coherent
account of partition bijections. [...]
This is a truly important contribution.
The author's presentation is lucid, often clarifying or improving the
original work and providing new insights."

Download .pdf
file of the paper and .pdf file
of the MathSciNet review.
Warning: The paper file is 660K. Printing on a monochromatic
printer may distort some colored pictures.

We present a geometric framework for a class of partition
identities. We show that there exists a unique bijection
proving these identities, and satisfies certain linearity
conditions. In particular, we show that Corteel's bijection
enumerating partitions with nonnegative k-th differences
can be obtained by our approach. Other examples and
generalizations are presented.

We show that the
Kauffman bracket
[L] of a checkerboard
colorable virtual link L is an evaluation of the
Bollobás-Riordan
polynomial RG of a
ribbon graph G = GL associated with L.
This result generalizes the celebrated relation between the
Kauffman bracket and the
Tutte polynomial
of planar graphs.

We give a number of new combinatorial interpretations of values of the
Tutte polynomial
of planar graphs, in terms of two different graph colorings,
claw coverings, and, for particular graphs on a square grid, in terms of
Wang tilings
and T-tetromino tilings.
These results are extended to surfaces of higher genus
and give interpretations of the Bollobás-Riordan polynomial.
Most proofs are bijective.

We present two versions of the paper which differ only
in the pictures. The first, colored version is for viewing
on a monitor and printing on a colored printer. The second,
monochromatic version, is optimized for printing on a
monochromatic printer.

We present a number of results enumerating spanning trees and spanning
forests in multipartite graphs and more general graphs with part structure.
This is an advanced extension of [Pak-Postnikov] and [Kelmans].

In this paper we extend the
loop-erased random walk (LERW) to
the directed hypergraph setting. We then generalize Wilson's algorithm
for uniform sampling of spanning trees to directed hypergraphs.
In several special cases, this algorithm perfectly samples spanning
hypertrees in expected polynomial time.

Our main application is to the reachability problem, also
known as the directed all-terminal network reliability problem.
This classical problem is known to be #P-complete, hence
is most likely intractable. We show that in the case of
bi-directed graphs, a conjectured polynomial bound for
the expected running time of the generalized Wilson's algorithm
implies a FPRAS for approximating reachability.

We study a long standing open problem on the mixing time
of Kac's
random walk on SO(n,R) by random rotations. We obtain
an upper bound mix =O(n2.5 log n) for the weak convergence
which is close to the trivial lower bound Ω (n2).
This improves the upper bound
O(n4 log n) by Diaconis and Saloff-Coste.
The proof is a variation on the coupling technique we
develop to bound the mixing time for compact Markov chains,
which is of independent interest.

Mixing time and long paths in graphs,
in Proc. SODA'02
(San Francisco, CA), 321-328.

We prove that regular graphs with large degree and small mixing
time contain long paths and other graphs. We apply the results to size
Ramsey numbers,
self-avoiding walks
in graphs, and present
efficient algorithm for finding long paths in graphs as above.

Let G be a group with
Kazhdan's
property (T), and let S be
a transitive generating set (there exists a subgroup H
of Aut(G) which acts transitively on S.) In this paper
we relate two definitions of the Kazhdan constant and the eigenvalue
gap in this case. Applications to various random walks on groups,
and the product replacement random algorithm, are also presented.

We investigate mixing of random walks on Sn and
An generated by permutations of a given cycle structure.
In our approach we follow methods developed by Diaconis,
by using characters of the symmetric group
and combinatorics of Young tableaux.
We conclude with conjectures and open problems.

We present an upper bound O(n2) for the mixing time of a simple
random walk on upper triangular matrices. We show that this bound is
sharp up to a constant, and find tight bounds on the eigenvalue gap.
We conclude by applying our results to indicate that the asymmetric
exclusion process on a circle indeed mixes more rapidly than the
corresponding symmetric process.

There are several examples where the mixing time of a
Markov chain
can be reduced substantially, often to about its square root, by
``lifting'', i.e., by splitting each state into several states. In
several examples of random walks on groups, the lifted chain not
only mixes better, but is easier to analyze.

We characterize the best mixing time achievable through lifting in
terms of
multicommodity flows.
We show that the reduction to square
root is best possible. If the lifted chain is time-reversible, then
the gain is smaller, at most a factor of log (1/p), where
p is the smallest stationary probability of any state. We give
an example showing that a gain of a factor of
log (1/p) log log (1/p) is possible.

Consider a random walk on a vector space with steps defined
by a given set of vectors. We show that in some cases the mixing time
can be defined in purely combinatorial terms. We also investigate
cutoff phenomenon for these walks.

We show that one can successfully employ stopping times to get sharp
bounds on mixing times for a wide range of examples of walks on
permutation and linear groups. The first half of the thesis dedicated
to a general theory of stopping times.

A classical conjecture by Graham Higman states that the number of conjugacy classes of Un(Fq), the group of upper triangular n×n matrices over Fq, is polynomial in q, for all n. In this paper we present both positive and negative evidence, verifying the conjecture for n≤16, and suggesting that it probably fails for n≥59. The tools are both theoretical and computational. We introduce a new framework for testing Higman's conjecture, which involves recurrence relations for the number of conjugacy classed of pattern groups. These relations are proved by the orbit method for finite nilpotent groups. Other applications are also discussed.

The classical
Lovász
conjecture says that every connected
Cayley graph
is Hamiltonian.
We present a short survey of
various results in that direction and make some additional
observations. In particular, we prove that every finite
group G has a generating set of size at most
log2 |G|,
such that the corresponding Cayley graph contains a
Hamiltonian cycle. We also present an explicit construction
of 3-regular Hamiltonian expanders.

(with Christopher Malon) Percolation on Finite Cayley
Graphs, Combinatorics, Probability and Computing,
vol. 15 (2006), 571-588.
Extended abstract of the earlier version
of the paper has appeared in
Proc. RANDOM'02.

In this paper, we study percolation on finite Cayley graphs.
A conjecture of Benjamini says that the critical percolation pc
of such a graph can be bounded away from one, for any Cayley
graph satisfying a certain diameter condition.
We prove Benjamini's conjecture for some special classes
of groups. We also establish a reduction theorem, which allows
us to build Cayley graphs for large groups without increasing
pc.

Download .pdf file of the paper.
Warning: Please do not use/refer to the RANDOM extended abstract as
the the results and proofs in the final version significantly differ.

The automorphism group of a free group Aut(Fk) acts on
the set of generating k-tuples
(g1,...,gk) of a
group G.
Higman
showed that when k=2, the union of conjugacy classes
of commutators [g1,g2] and
[g1,g2]-1 is an orbit invariant.
We give a negative answer to a question of
B.H. Neumann,
as to whether there is a generalization of Higman's result for k > 2.

On probability of generating a finite group,
preprint, 1999; later largely included into "What do we know..." survey article.

Let G be a finite group, and let p(G,k) be the
probability that k random
group elements generate G. Denote by v(G) the smallest
k such that p(G,k)>1/e. In this paper we analyze
the quantity v(G) for different classes of groups.
We prove that v(G)< r(G)+1 when G is
nilpotent,
and r(G) is the minimal number of generators of G.
When G is solvable we show that
v(G) < 3.25 r(G) + 107.
We also show that v(G) < C log log |G|,
where G is a direct product of simple nonabelian groups
and C is a universal constant. Applications to the
"product replacement algorithm" are also discussed.

It is known that random 2-lifts of graphs give rise to expander graphs. We present a new conjectured derandomization of this construction based on certain Mealy automata. We verify that these graphs have polylogarithmic diameter, and present a class of automata for which the same is true. However, we also show that some automata in this class do not give rise to expander graphs.

We prove the exponential growth of product replacement graphs
for a large class of groups. Much of our effort is dedicated to
the study of product replacement graphs of Grigorchuk groups,
where the problem is most difficult.

We construct an uncountable family of finitely generated groups of
intermediate growth, with growth functions of new type.
These functions can have large oscillations between lower and
upper bounds, both of which come from a wide class of functions.
In particular, we can have growth oscillating between exp(nα)
and any prescribed function, growing as rapidly as desired.
Our construction is built on top of any of the Grigorchuk
groups of intermediate growth, and is a variation on the
limit of permutational wreath product.

We present an accessible introduction to basic results on
groups of intermediate
growth.
The idea is to separate the (straightforward) analytic calculations and group
theoretic arguments. We neither survey the field nor
do we obtain the best known bounds for the growth.
Instead, we concentrate on making the exposition as
elementary and self-contained as possible.

For every
nonamenable group,
a finite system of generators is constructed
such that the Bernoulli
bond percolation
on the corresponding Cayley graph exhibits the double phase transition
phenomenon, i.e., nonempty nonuniqueness phase.

We present an analytic technique for estimating the
growth for groups of intermediate growth.
We apply our technique to
Grigorchuk groups,
which are the only known examples of such groups. Our estimates
generalize and improve various bounds by Grigorchuk, Bartholdi and
others.

Let pc(G) be the critical probability of the site percolation
on the Cayley graph of group G. Benjamini and Schramm
conjectured that pc<1, given the group is infinite
and not a finite
extension of Z. The conjecture was proved earlier for groups of
polynomial and exponential growth and remains open for groups of
intermediate growth. In this note we prove the conjecture for a
special class of
Grigorchuk groups,
which contains all known examples of groups of intermediate growth.

We establish a connection between the
expansion coefficient of the product
replacement graph of a group G, and the minimal expansion
coefficient of a
Cayley graph of G.
This gives a new explanation of the outstanding performance of
the product replacement algorithm and supports the speculation
that all product replacement graphs are
expanders.

The main result of this paper is a polynomial upper bound
for the cost of the algorithm, provided k is large enough.
This is the first such result, improving (sub)-exponential bounds
by Diaconis and
Saloff-Coste, etc.

The ``product replacement algorithm'' is a commonly used
heuristic to generate random group elements in a finite group G,
by running a random walk on generating k-tuples of G.
While experiments showed outstanding performance, the theoretical
explanation remained mysterious.
In this paper we propose a new approach to study of the algorithm,
by using
Kazhdan's property (T)
from representation theory of
Lie groups.

We prove that the product replacement graph
on generating 3-tuples of An is connected for n < 12.
We employ an efficient heuristic based on the ``large connected component''
concept and use of symmetry to prune the search. The heuristic works for any
group. Our tests were confined to An due to the interest in
Wiegold's Conjecture, usually stated in terms of T-systems.
Our results confirm Wiegold's Conjecture in some special cases and
are related to the recent conjecture of Diaconis and Graham.
The work was motivated by the study of
the product replacement algorithm.

What do we know about the product replacement algorithm?,
Groups and Computation III (W. Kantor, A. Seress, eds.),
de Gruyter, Berlin, 2001, 301-347.

We give an extensive review of the theoretical results related
to the product replacement algorithm. Both positive and negative
results are described.
The review is based on a large amount of work done by the author,
including joint results with Babai, Bratus, Cooperman, Lubotzky
and Zuk (see on this web page).

(with Lászl&oacute Babai)
Strong bias of group generators: an
obstacle to the ''product replacement algorithm'',
Journal of Algorithms, vol. 50 (2004), 215-231
(special SODA'2000 issue).
An extended abstract of this paper has appeared in
Proc. SODA'00, 627-635.

Let G be a finite group. Efficient generation of nearly
uniformly distributed
random elements in G, starting from a given set of generators
of G, is a central problem in computational group theory. In this
paper we demonstrate a weakness in the popular ''product replacement
algorithm,'' widely used for this purpose. Roughly, we show that
components of the uniform generating k-tuples have a bias
in the distribution, detectable by a short straight-line program.

On the graph of generating sets of a simple group,
preprint, 1999; later largely incorporated into "What do we know..." survey.

We prove that the product replacement graph
on generating k-tuples of a simple group contains a
large connected component. This is related to the recent
conjecture of
Diaconis
and Graham.
As an application, we also prove
that the output of the product replacement algorithm
in this case does not have a strong bias.

This is an extended abstract of the two separate papers on the
generating k-subsets of a finite group. We elaborate on
the number of such subsets and present an efficient and very
economic algorithm in case of
nilpotent groups.
We also prove rapid mixing of the product replacement algorithm in case when
group is abelian.

Testing commutativity of a group and the power of randomization, LMS Journal of Computation and Mathematics,
vol. 15 (2012), 38-43.

Let G be a group generated by k elements,
with group operations (multiplication, inversion, comparison with identity)
performed by a black box. We prove that one can test whether G
is abelian
at a cost of O(k) group operations.
On the other hand, we show that deterministic approach requires
Ω(k2) group operations.

This paper first appeared as a Yale preprint in 2000.
It was revised and updated in 2011 prior to submission.
Download .pdf file of the paper.
See here
for a journal version.

Let G be a finite group. For a given k, what is the probability
that a group is generated by k random group element? How small
can be this probability and how one can uniformly sample these
generating k-tuples of elements? In this paper we answer
these questions for
nilpotent and
solvable groups.
Applications to product replacement algorithms and random random walks are
discussed.

We present a Las Vegas algorithm
for verification whether a given group defined as a black box group (we can multiply
elements, take inverses, and compare them with identity) is
isomorphic to a given
symmetric group.
Surprisingly, the algorithm relies on the
Goldbach conjecture
and its various extensions.
In the appendix to the article we use analytic number theory
and probabilistic approach to support the conjectures.

We give a combinatorial proof of the inequality in the title in terms of
Fibonacci numbers
and Euler numbers.
The result is motivated by Sidorenko's theorem
on the number of linear extensions of the poset and its complement. We conclude
with some open problems.

Download .pdf file of the paper. Here is a short
blog post on the paper.
Warning: The picture on the front page looks best when printed in color.

We give a brief history of Catalan numbers, from their first discovery in the
18th century to modern times.

From Tina Garrett's review of the book:
[..] the appendices contain a must-read history of the Catalan numbers by Igor Pak. In
particular, Pak’s description of the early years of Catalan research, from Euler’s communication
with Segner to the 19th century work of several French mathematicians,
takes the reader on a journey of discovery.
Igor Pak’s contribution, History
of Catalan Numbers is a delightful look at the emergence of this fascinating sequence
in the historical literature and will be of interest to those who think deeply about how
mathematical discovery emerges.

Download .pdf file of the paper.
See my Catalan Numbers website with historical documents and other links.
See also my blog posts on which this paper was based: one and two.

We present combinatorial proofs of several
Fine's partition
theorems, along with some historical account.

Download .pdf file of the paper.
Download here the journal version of the paper (note the bio sketch at the end).

A preliminary version of this paper was translated and
published in
Matematicheskoe
Prosveschenie, vol. 7 (2003), 136-149 (in Russian).
This is an annual publication of MCCME.
Download the .pdf file
of the Russian translation here
or here.